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EXERCISE 1.1 Q: 1

Is zero a rational number? Can you write it in the form , where p and q are integers and q 0? Answer

Yes zero is a rational number as it can be represented in the form, where p and q are

integers and q 0 as etc. Concept Insight: Key idea to answer this question is "every integer is a rational number and

zero is a non negative integer". Also 0 can be expressed in form in various ways as 0 divided

by any number is 0. simplest is .

Q: 2 Find six rational numbers between 3 and 4. Answer There are infinite rational numbers in between 3 and 4.

3 and 4 can be represented as respectively. Now rational numbers between 3 and 4 are

Concept Insight: Since there are infinite number of rational numbers between any two numbers

so the answer is not unique here. The trick is to convert the number to equivalent form by multiplying and dividing by the number atleast 1 more than the rational numbers to be inserted.

Q: 3

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Find five rational numbers between . Answer

There are infinite rational numbers between

Now rational numbers between are

Concept Insight: Since there are infinite number of rational numbers between any two numbers

so the answer is not unique here. The trick is to convert the number to equivalent form by multiplying and dividing by the number at least 1 more than the rational numbers required.

Alternatively for any two rational numbers a and b, is also a rational number which lies between a and b.

Q: 4 State whether the following statements are true or false. Give reasons for your answers. (i) Every natural number is a whole number (ii) Every integer is a whole number (iii) Every rational number is a whole number

Answer (i) True, since collection of whole numbers contains all natural numbers. (ii) False, integers include negative of natural numbers as well which are clearly not whole numbers. For example -1 is an integer but not a whole number

(iii) False, rational numbers includes fractions and integers as well. For example is a rational

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number but not whole number. Concept Insight: Key concept involved in this question is the hierarchy of number systems

Remember the bigger set consists of the smaller one. Since Mathematics is an exact science every fact has a proof but in order to negate a statement even on e counter example is sufficient.

EXERCISE 1.2

Q: 1 State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number.

(ii) Every point on the number line is of the form , where m is a natural number. (iii) Every real number is an irrational number. Answer (i) True, since real numbers consists of rational and irrational numbers.

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(ii) False, Since negative integers cannot be expressed as the square root of any natural number. (iii) False, real number includes both rational and irrational numbers. So every real number can not be an irrational number. Concept Insight: Mentioning the reasons is important in this problem. Real Numbers consists of rational and irrational numbers and not vice versa. Every real number corresponds to a point on number line and vice versa. Recall real number includes negative numbers also. Square root of negative numbers is not defined.

Square roots of all square numbers is rational.

For example Thus the square roots of all positive integers are not irrational Concept Insight: In general only the square root of a prime number is irrational. There are the perfect square numbers.

Q: 3

Q: 2 Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number. Answer

Show how can be represented on the number line. Answer Using Pythagoras Theorem: 5=22+12

Taking positive square root we get

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1. Mark a point 'A' representing 2 units on number line. 2. Now construct AB of unit length perpendicular to OA. Join OB 3. Now taking O as centre and OB as radius draw an arc, intersecting number line at point C.

4. Point C represents on number line

Concept Insight: For a positive integer n, can be located on number line , if is located using Pythagoras Theorem . If is a perfect square then this method is useful.

To represent the irrational number key idea is to use Pythagoras theorem and create a

length of units by constructing a right triangle of base and perpendicular of length 2 and 1 units.

EXERCISE 1.3

Q: 1 Write the following in decimal form and say what kind of decimal expansion each has:

(i) (ii) (iii)

(iv) (v) (vi)

Answer

(i)

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terminating

(ii) non terminating repeating

(iii) Terminating

(iv) non terminating repeating

(v) non terminating repeating decimal

(vi) Terminating decimal Concept Insight: The decimal expansion of a rational number is either terminating or non terminating recurring. Decimal expansion terminates in case the prime factors of denominator includes 2 or 5 only.

Q: 2

You know that . Can you predict what the decimal expansion of

are, without actually doing the long division? If so, how? Answer Yes it can be done as follows:

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Concept Insight: Multiples of the given decimal expansion can be obtained by simple multiplication with the given constant. Cross check the answer by performing long division.

Q: 3

Express the following in the form , where p and q are integers and .

Answer (i) Let x = 0.666 ... (i) Multiplying by 10 we get 10x = 6.666 ... (ii) (ii) - (i) gives 9x = 6

Or x =

(ii) Let x = 0.4777 ... (i) 10x = 4.777 ... 100x = 47.777 ... (ii) (ii) - (i) gives 99 x = 43

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x =

(iii) Let x = 0.001001 ...(i) 1000x = 1.001001 ...(ii) (ii) - (i) gives 999x = 1

x = Concept Insight: The key idea to express a recurring decimal in the p/q form is to multiply the number by the 10n where n = number of digits repeating. This is done to make the repeating block a whole number part of the decimal. By subtracting the two expressions x can be expressed in the P/q form

Q: 4

Express 0.99999 ..... in the form . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense. Answer Let x = 0.9999 .. .. ..(i) 10x = 9.9999 ... ...(ii) (ii) - (i) gives 9x = 9 x = 1 Concept Insight: .9999999 ..... is nothing but 1 when expressed in p/q form.

Q: 5 What can be the maximum number of digits be in the repeating block of digits in the decimal

expansion of ? Perform the division to check your answer. Answer

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Expressing in the decimal form we

There are 16 digits in repeating block of decimal expansion of . Concept Insight: Maximum number of digits that can repeat will be 1 less than the prime number in denominator.

Q: 6

Look at several examples of rational numbers in the form where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy? Answer

Terminating decimal expansion will come when denominator q of rational number , is either of 2, 4, 5, 8, 10, and so on ... ...

Terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions are having power of 2 only or 5 only or both. Concept Insight: A rational number in its simplest form will terminate only when prime factors of its denominator consists of 2 or 5 only.

Q: 7 Write three numbers whose decimal expansions are non-terminating non-recurring.

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Answer 3 numbers whose decimal expansion is non terminating non recurring are ... ... , 0.505005000051509 ... ... ... 0.72012009200011500007200000 ... ... ... 7.03124509761202 ... ... ... ... ... ... Concept Insight: Recall that a non terminating non recurring decimal is an irrational number. Answer to such questions is not unique.

Q: 8

Find three different irrational numbers between the rational numbers Answer

3 irrational numbers are - 0.73073007300073000073 ... ... ... 0.75075007500075000075 ... ... ... 0.79079007900079000079 ... ... ...

Concept Insight: There is infinite number of rational and irrational numbers between any two rational numbers. Convert the number into its decimal form to find irrationals between them.

Alternatively following result can be used to answer

Irrational number between two numbers x and y

Q: 9 Classify the following numbers as rational or irrational:

Answer

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(i) As decimal expansion of this number is non-terminating non recurring. So it is an irrational number.

(ii)

Rational number as it can be represented in form. (iii) 0.3796 As decimal expansion of this number is terminating, so it is a rational number. (iv) As decimal expansion of this number is non terminating recurring so it is a rational number. (v) As decimal expansion of this number is non terminating non repeating so it is an irrational number. Concept Insight: A number is rational if its decimal expansion is either terminating or non terminating but recurring. A number which cannot be expressed in p/q form is irrational. Square root of prime numbers is always irrational.

EXERCISE 1.4

Q: 1 Visualise 3.765 on the number line using successive magnification. Answer 3.765 can be represented

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Concept Insight: Divide the number line between the number to be represented in 10 parts starting the whole number part.

Q: 2

Visualise on the number line, up to 4 decimal places. Answer

= 4.2626

We can visualise 4.2626 as in following steps

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Concept Insight: Divide the number line between the number to be represented in 10 parts starting the whole number part.

EXERCISE 1.5

Q: 1 Classify the following numbers as rational or irrational:

Answer

(i) As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.

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(ii)

It can be represented in form so it is a rational number.

(iii)

As it can be represented in form, so it is a rational number.

(iv) As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.

(v) As decimal expansion is non terminating non recurring, so it is an irrational number. Concept Insight: Do the simplifications as indicated and see whether the number is terminating, non terminating recurring or neither terminating nor repeating. Remember Sum/difference/Product of a rational and irrational number may or may not be irrational.

Q: 2 Simplify each of the following expressions:

Answer

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Concept Insight: Apply the algebraic identities (a+b)2, (a-b)2

Q: 3

,(a+b)(a-b) etc to simplify the given expressions. Equivalent Identities used here are

Recall, is defined as the ratio of the circumference (say c) of a circle to its diameter (say d).

That is, . This seems to contradict the fact that is irrational. How will you resolve this contradiction? Answer

There is no contradiction. Since here circumference or diameter are not given to be integers . When we measure a length with scale or any other instrument, we only get an approximate rational value. We never get an exact value. c or d may be irrational. So, the

fraction is irrational. Therefore, is irrational.

Concept Insight: A rational number is the number of the form where p and q are

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integers. In c and d are not integers. Also remember that no measurement is exact. Q: 4

Represent on the number line. Answer (i) Mark a line segment OB = 9.3 on number line. (ii) Take BC of 1 unit. (iii) Find mid point D of OC and draw a semicircle on OC while taking D as its centre. (iv) Draw a perpendicular to line OC passing through point B. Let it intersect semicircle at E.

Length of perpendicular BE = . (v) Taking B as centre and BE as radius draw an arc intersecting number line at F. BF

is i.e point F represents on number line

Verification: In EDB ED2=EB2+DB2

Using Pythagoras theorem

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Concept Insight: This method based on the application of Pythagoras theorem can be used to represent root of any rational number on the number line.

The key idea to represent is to create a length of units.

In ODB

DB = Q: 5 Rationalise the denominators of the following:

Answer

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Concept Insight: Rationalisation of denominator means converting the irrational denominator

to rational i.e . removing the radical sign from denominator.A number of the form can be converted to rational form by multiplying with its conjugate. Remember the algebraic identities

EXERCISE 1.6 Q: 1

Answer

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Concept Insight: Express the number in exponent notation and use the rule Exponent m must be such that it is divisible by n. Q: 2

Answer

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Concept Insight: Express the number in exponent notation and use the rule of exponents. Q: 3

Answer

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Concept Insight: Use the rule of exponents